This invention relates to a method of distributing dimples on a golf ball utilizing principles of electromagnetic field theory.
One of the most fundamental equations in engineering mathematics is Laplace""s Equation. A number of physical phenomena are described by this partial differential equation including steady-state heat conduction, incompressible fluid flow, elastostatics, as well as gravitational and electromagnetic fields. The theory of solutions of this equation is called potential theory.
One example of potential theory is electromagnetic field theory, which can be used to distribute objects on a spherical surface. Electromagnetic field theory has been studied extensively over the years for a variety of applications. It has been used, for example, in satellite mirror design. Electromagnetic field theory, including the obvious applications to semiconductor research and computer technology, has many applications in the physical sciences, not limited to celestial mechanics, organic chemistry, geophysics, and structural acoustics.
In many applications, the objects are treated as point charges so that principles of electromagnetic field theory can be applied to determine optimal positioning or to predict the equilibrium positions of the objects.
While the task of distributing point charges on a spherical surface has been studied extensively in mathematical circles, it has not been employed as a tool to develop and define dimple patterns or optimal dimple distributions on a golf ball.
Instead, current golf ball dimple patterns generally are based upon dividing the spherical surface of the ball into discrete polygonal surfaces. The edges of the surfaces join to form geometric shapes that approximate the spherical surface of a golf ball. These geometric shapes include, for example, regular octahedral, regular icosahedral and regular polyhedral arrangements. Once a geometric shape is selected, the polyhedral surfaces are individually filled with a dimple pattern that may be repeated over the surface.
While this approach may be effective in enabling easy dimple design and mold manufacture, it may not result in optimal dimple positioning or distribution for improved aerodynamic performance. In addition, this approach to designing a dimple pattern may result in a golf ball having variations in flight performance depending upon the direction of rotation of the ball. For instance, rotation about one axis may result in different flight characteristics than rotation about a second axis. Moreover, the difference may be large enough to produce a measurable and visible difference in aerodynamic lift and drag.
The potential limitations described above may be present in other methods for arranging dimples on a golf ball. Thus, it would be desirable to have a way to optimize a dimple pattern by repositioning the dimples to improve flight performance.
The present invention uses electromagnetic field theory implemented as a numerical computer algorithm to create dimple patterns and to optimize dimple placement and distribution on a golf ball. The method solves the constrained optimization problem where the objective function is an electric potential function subject to various constraints, such as dimple spacing or size. A number of potential functions can be utilized to describe the point charge interactions. A variety of optimization methods are available to minimize the objective function including gradient based, response surface, and neural network algorithms. These solution strategies are readily available and known to one skilled in the art. One embodiment of the present invention uses a Coulomb potential function and a gradient based solution strategy to create a dimple pattern.
One benefit from using these principles to develop dimple patterns is that doing so may result in a golf ball having improved aerodynamic performance.
Use of the inventive method provides a golf ball having a plurality of dimples on its surface, some of which have been positioned on the golf ball surface according to principles of electromagnetic theory. At first, the dimples that are to be positioned according to these principles may be randomly distributed on at least a portion of the golf ball surface. The ball surface may be divided into hemispheres, quadrants, or according to platonic solid shapes in order to define the portion of the golf ball on which the dimples will be arranged.
In one embodiment, the dimples are placed on a hemispherical portion of the golf ball. In another embodiment, the dimples are placed on the entire spherical portion of the ball. In yet another embodiment, the dimples are placed on the regions defined by an Archimedean solid, most preferably a great rhombicosidodecahedron.
The dimples may have any desired shape, although in a preferred embodiment the dimples are circular. In another embodiment, however, the dimples are polygonal in shape. In addition the dimples may be of any desired number. In one embodiment, the dimples are between about 200 to about 600 in number. In a preferred embodiment, the dimples are between about 300 to about 500 in number.
The size of the dimples may also vary. In one embodiment, the dimples are between about 0.04 to about 0.1 inches when measured from the centroid of the dimple to its outermost extremity. More preferably, the dimples are about 0.05 to about 0.09 inches in size. In yet another embodiment, the dimples are substantially circular and have varying diameters sizes from about 0.04 to about 0.20 inches, and more preferably are between about 0.100 and about 0.180 inches.
In general, the present invention involves a method for optimizing the arrangement of dimples on a golf ball under the principles developed by potential theory. In one embodiment, the steps of the method include defining a region or portion of the ball surface in which dimples will be arranged, placing dimples within the defined region or portion of the ball, and assigning charge values to each dimple. The potential of the charges are determined and a solution method is applied to minimize the potential. In a preferred embodiment, the solution method used is gradient-based. The solution method allows the dimples to be rearranged or altered and the steps repeated until the potential has reached a predetermined tolerance or has been sufficiently minimized. In a preferred embodiment, the steps are repeated until the gradient is approximately zero.
In one embodiment, at least one dimple is substantially circular, while in another embodiment a plurality of dimples are circular and have diameters from about 0.05 to about 0.200 inches. In yet another embodiment, at least one additional dimple is placed on the ball surface outside of the defined portion of the golf ball.
Some of the dimples arranged on the surface of a golf ball under the present invention may have any desired plane shape. The dimples may be, for instance, circular, oval, triangular, rhombic, rectangular, pentagonal, polygonal, or star shaped. The present invention is not limited to any minimum or maximum number of dimples that may be used, but in a preferred embodiment the total number of dimples on the golf ball is from about 200 to about 1000 dimples, and an even more preferred total number of dimples is from about 200 to about 600 dimples.
One embodiment further comprises the step of defining a portion of the ball where dimples will not be arranged. For example, it is preferred that no dimple is placed across a mold plate parting line. In yet another embodiment, the defined portion of the golf ball surface is from about one-eighth to about one half of a hemisphere of the ball surface, and more preferably corresponds to approximately one-fifth of a the ball""s surface. The optimized dimple arrangement with these defined regions may be repeated on additional portions of the golf ball.
In one embodiment of the present invention the completed dimple pattern has at least about 74 percent dimple coverage, while it is preferred that the dimple surface coverage is at least about 77 percent. In another embodiment the completed dimple pattern has about 82 percent or greater surface coverage.